Calculus Answers

Question 2

(a) Given that A ~ = (x + 2y + 4z)ˆ

i + (2x - 3y- z)ˆj +(4x-y + 2z)kˆ

(i). Show that the vector field A ~ is irrotational

(ii). Find the scalar potential φ such that A~ = ∇φ, if φ(0, 0, 0) = 1

Show that ∇𝑟ⁿ = 𝑛𝑟^n-2"\\vec r"

Question:

If 𝜑 = ln"|\\vec r|"

, then show that ∇𝜑 = "\\vec r" /𝑟2

Find the Maclaurin series or Taylor series centered at a, interval of

convergence and the radius of convergence of 1

√1−𝑥

.

For the Fourier series in Problem 1, deduce a

series for

π

4

at the point where x =

π

2

. Determine the Fourier series for the periodic

function:

f (x) =  −

2

,

when

−π <

x

< 0

+

2

,

when 0

< x

< π

which is periodic outside this range of

period 2π.

Evaluate ∫ ∫ ∫ X 2YZ D X D Y D Z Over the Volume Bounded by Planes X=0,Y=0, Z=0 and X + Y + Z = 1